Cubic exponential integral Stokes sectors ($G(z)={\int }_0^1{e}^{-z^3{t}^3}dt$)

The saddle and endpoint asymptotic expansions of this entire function have algebraic coefficients $C{\omega }^j/z$ in sectors $(2j-1)\pi /3<\arg z<(2j+1)\pi /3$, where $C=\Gamma (1/3)/3$ and $\omega =e^{2\pi i/3}$. Deforming the contour onto the decay ray $\arg (zt)=2j\pi /3$ gives its complete integral $C{\omega }^j$. The endpoint contribution is $-e^{-z^3}(3z^3{)}^{-1}{\sum }_{r\ge 0}(-1{)}^r(2/3{)}_r/z^{3r}$. The algebraic coefficient switches at $\arg z=\pi /3,\pi ,5\pi /3$; the equal-exponential-magnitude anti-Stokes lines are the intervening rays $\pi /6+j\pi /3$.

Near $\arg z=\pi /3$, set $\lambda =-z^3$ on the branch with argument near zero and truncate the endpoint expansion at $N$ terms. Its remaining factorial series, indexed by $r=N+p$, is ${\lambda }^{-1/3}/[3\Gamma (2/3)]$ times the Borel summation of ${\sum }_{p\ge 0}\Gamma (N+2/3+p)e^{\lambda }/{\lambda }^{N+2/3+p}$. The upper-pole contour fixes the stated remainder and its continuation from the sector below the switching line. With $N+2/3\sim |z{|}^3$, the Gaussian pole transition for a large gamma parameter smooths the algebraic coefficient from $C$ to $C{e}^{2\pi i/3}$.